Question: Simplify the following expression and state the condition under which the simplification is valid. $t = \dfrac{z^3 - 11z^2 + 10z}{-4z^3 + 12z^2 + 280z}$
Explanation: First factor out the greatest common factors in the numerator and in the denominator. $ t = \dfrac {z(z^2 - 11z + 10)} {-4z(z^2 - 3z - 70)} $ $ t = -\dfrac{z}{4z} \cdot \dfrac{z^2 - 11z + 10}{z^2 - 3z - 70} $ Simplify: $ t = - \dfrac{1}{4} \cdot \dfrac{z^2 - 11z + 10}{z^2 - 3z - 70}$ Since we are dividing by $z$ , we must remember that $z \neq 0$ Next factor the numerator and denominator. $ t = - \dfrac{1}{4} \cdot \dfrac{(z - 10)(z - 1)}{(z - 10)(z + 7)}$ Assuming $z \neq 10$ , we can cancel the $z - 10$ $ t = - \dfrac{1}{4} \cdot \dfrac{z - 1}{z + 7}$ Therefore: $ t = \dfrac{ -z + 1 }{ 4(z + 7)}$, $z \neq 10$, $z \neq 0$